3.1016 \(\int x^{15} \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{11/4}}{11 b^4}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^4}-\frac{a \left (a+b x^4\right )^{15/4}}{5 b^4} \]

[Out]

-(a^3*(a + b*x^4)^(7/4))/(7*b^4) + (3*a^2*(a + b*x^4)^(11/4))/(11*b^4) - (a*(a +
b*x^4)^(15/4))/(5*b^4) + (a + b*x^4)^(19/4)/(19*b^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.106935, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^3 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{11/4}}{11 b^4}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^4}-\frac{a \left (a+b x^4\right )^{15/4}}{5 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^15*(a + b*x^4)^(3/4),x]

[Out]

-(a^3*(a + b*x^4)^(7/4))/(7*b^4) + (3*a^2*(a + b*x^4)^(11/4))/(11*b^4) - (a*(a +
b*x^4)^(15/4))/(5*b^4) + (a + b*x^4)^(19/4)/(19*b^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.2948, size = 70, normalized size = 0.88 \[ - \frac{a^{3} \left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{4}} + \frac{3 a^{2} \left (a + b x^{4}\right )^{\frac{11}{4}}}{11 b^{4}} - \frac{a \left (a + b x^{4}\right )^{\frac{15}{4}}}{5 b^{4}} + \frac{\left (a + b x^{4}\right )^{\frac{19}{4}}}{19 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15*(b*x**4+a)**(3/4),x)

[Out]

-a**3*(a + b*x**4)**(7/4)/(7*b**4) + 3*a**2*(a + b*x**4)**(11/4)/(11*b**4) - a*(
a + b*x**4)**(15/4)/(5*b**4) + (a + b*x**4)**(19/4)/(19*b**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0340574, size = 61, normalized size = 0.76 \[ \frac{\left (a+b x^4\right )^{3/4} \left (-128 a^4+96 a^3 b x^4-84 a^2 b^2 x^8+77 a b^3 x^{12}+385 b^4 x^{16}\right )}{7315 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^15*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(3/4)*(-128*a^4 + 96*a^3*b*x^4 - 84*a^2*b^2*x^8 + 77*a*b^3*x^12 + 3
85*b^4*x^16))/(7315*b^4)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 47, normalized size = 0.6 \[ -{\frac{-385\,{b}^{3}{x}^{12}+308\,a{b}^{2}{x}^{8}-224\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{7315\,{b}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15*(b*x^4+a)^(3/4),x)

[Out]

-1/7315*(b*x^4+a)^(7/4)*(-385*b^3*x^12+308*a*b^2*x^8-224*a^2*b*x^4+128*a^3)/b^4

_______________________________________________________________________________________

Maxima [A]  time = 1.42589, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{19}{4}}}{19 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}} a}{5 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2}}{11 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^15,x, algorithm="maxima")

[Out]

1/19*(b*x^4 + a)^(19/4)/b^4 - 1/5*(b*x^4 + a)^(15/4)*a/b^4 + 3/11*(b*x^4 + a)^(1
1/4)*a^2/b^4 - 1/7*(b*x^4 + a)^(7/4)*a^3/b^4

_______________________________________________________________________________________

Fricas [A]  time = 0.369758, size = 77, normalized size = 0.96 \[ \frac{{\left (385 \, b^{4} x^{16} + 77 \, a b^{3} x^{12} - 84 \, a^{2} b^{2} x^{8} + 96 \, a^{3} b x^{4} - 128 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{7315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^15,x, algorithm="fricas")

[Out]

1/7315*(385*b^4*x^16 + 77*a*b^3*x^12 - 84*a^2*b^2*x^8 + 96*a^3*b*x^4 - 128*a^4)*
(b*x^4 + a)^(3/4)/b^4

_______________________________________________________________________________________

Sympy [A]  time = 66.5883, size = 110, normalized size = 1.38 \[ \begin{cases} - \frac{128 a^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{4}} + \frac{96 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{3}} - \frac{12 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1045 b^{2}} + \frac{a x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{95 b} + \frac{x^{16} \left (a + b x^{4}\right )^{\frac{3}{4}}}{19} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} x^{16}}{16} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**15*(b*x**4+a)**(3/4),x)

[Out]

Piecewise((-128*a**4*(a + b*x**4)**(3/4)/(7315*b**4) + 96*a**3*x**4*(a + b*x**4)
**(3/4)/(7315*b**3) - 12*a**2*x**8*(a + b*x**4)**(3/4)/(1045*b**2) + a*x**12*(a
+ b*x**4)**(3/4)/(95*b) + x**16*(a + b*x**4)**(3/4)/19, Ne(b, 0)), (a**(3/4)*x**
16/16, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216494, size = 77, normalized size = 0.96 \[ \frac{385 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} - 1463 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a + 1995 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2} - 1045 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^15,x, algorithm="giac")

[Out]

1/7315*(385*(b*x^4 + a)^(19/4) - 1463*(b*x^4 + a)^(15/4)*a + 1995*(b*x^4 + a)^(1
1/4)*a^2 - 1045*(b*x^4 + a)^(7/4)*a^3)/b^4